Puzzle ZUKB 

Workings

Hint
There are no larger squares than size 4x4.

Answer
There are 110 squares without a rook.

Reasoning
There are 60 squares of size 1x1.
There are 35 squares of size 2x2.
There are 12 squares of size 3x3.
There are  3 squares of size 4x4.

Giving a total of 60 + 35 + 12 + 3 = 110 squares.